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Stability and boundedness results for solutions of certain third order nonlinear vector differential equations. (English) Zbl 1132.34328

Consider the nonlinear vector differential equation

x +F(x,x ˙,x ¨)x ¨+B(t)x ˙+h(x ˙)=p(t,x,x ˙,x ¨),t0,(*)

where F and B are symmetric n×n-matrices depending continuously on their arguments, the functions h and p mapping into n are also continuous functions. Using Lyapunov’s direct method, the authors present conditions guaranteeing the asymptotic stability of the zero solution of (*) and the boundedness of all solutions.

MSC:
34D20Stability of ODE
34C11Qualitative theory of solutions of ODE: growth, boundedness
References:
[1]Reissig, R., Sansone, G., and Conti, R., Nonlinear Differential Equations of Higher Order, Noordhoff, Groningen, 1974.
[2]Ezeilo, J. O. C., ’Periodic solutions of certain third order differential equations’, Atti della Accademia Nazionale dei Lincei. Rendiconti. Classe di Scienze Fisiche, Matematiche e Naturali. Serie 57(8), 1974, 1–2; 1975, 54–60.
[3]Ezeilo, J. O. C., ’Periodic solutions of certain third order differential equations of the nondissipative type’, Atti della Accademia Nazionale dei Lincei. Rendiconti. Classe di Scienze Fisiche, Matematiche e Naturali. Serie 63(8), 1977, 3–4; 1978, 212–224.
[4]Feng, C. H., ’On the existence of almost periodic solutions of nonlinear third-order differential equations’, Annals of Differential Equations 9(4), 1993, 420–424.
[5]Feng, C. H., ’The existence of periodic solutions for a third-order nonlinear differential equation’, (Chinese) Gongcheng Shuxue Xuebao 11(2), 1994, 113–117.
[6]Mehri, B., ’Periodic solution for certain nonlinear third order differential equations’, Indian Journal of Pure and Applied Mathematics 21(3), 1990, 203–210.
[7]Mehri, B. and Shadman, D., ’Periodic solutions of certain third order nonlinear differential equations’, Studia Scientiarum Mathematicarum Hungarica 33(4), 1997, 345–350.
[8]Qian, C., ’On global stability of third-order nonlinear differential equations’, Nonlinear Analysis: Theory, Methods & Applications Ser. A 42(4), 2000, 651–661.
[9]Qian, C., ’Asymptotic behavior of a third-order nonlinear differential equation’, Journal of Mathematical Analysis and Applications 284(1), 2003, 191–205. · Zbl 1054.34078 · doi:10.1016/S0022-247X(03)00302-0
[10]Shadman, D. and Mehri, B., ’On the periodic solutions of certain nonlinear third order differential equations’, Zeitschrift für Angewandte Mathematik und Mechanik 75(2), 1995, 164–166. · Zbl 0828.34029 · doi:10.1002/zamm.19950750218
[11]Tejumola, H. O., ’Periodic solutions of certain third order differential equations’, Atti della Accademia Nazionale dei Lincei. Rendiconti. Classe di Scienze Fisiche, Matematiche e Naturali. Serie 66(8), 1979, no. 4, 243–249.
[12]Vladimir, V., ’To existence of the periodic solution of a third-order nonlinear differential equation’, Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica 29, 1990, 123–163.
[13]Yoshizawa, T., ’Stability theory and the existence of periodic solutions and almost periodic solutions’, Applied Mathematical Sciences, Vol. 14, Springer-Verlag, New York-Heidelberg, 1975.
[14]Zhang, X. Z., ’Stability in the large of a class of third-order nonlinear ordinary differential equations’, Kexue Tongbao (English edn.) 28(7), 1983, 999–1000.
[15]Wang, L. and Wang, M. Q., ’On the construction of globally asymptotically stable Lyapunov functions for a type of nonlinear third-order system’, Acta Mathematicae Applicatae Sinica 6(3), 1983, 309–323.
[16]Abou-El-Ela, A. M. A., ’Boundedness of the solutions of certain third-order vector differential equations’, Annals of Differential Equations 1(2), 1985, 127–139.
[17]Afuwape, A.U., ’Ultimate boundedness results for a certain system of third-order nonlinear differential equations’, Journal of Mathematical Analysis and Applications 97(1), 1983, 140–150. · Zbl 0537.34031 · doi:10.1016/0022-247X(83)90243-3
[18]Afuwape, A. U., ’ Further ultimate boundedness results for a third-order nonlinear system of differential equations’, Unione Matematica Italiana Bollettino C. Serie VI 4(1), 1985, 347–361.
[19]Edziwy Spolhk, S., ’Stability of certain third order vector differential equations’, Bulletin de l’Académie Polonaise des Sciences. Série des Sciences Mathématiques, Astronomiques et Physiques 23, 1975, 15–21.
[20]Ezeilo, J. O. C. and Tejumola, H. O., ’Boundedness and periodicity of solutions of a certain system of third-order non-linear differential equations’, Annali di Matematica Pura ed Applicata 74(4), 1966, 283–316. · Zbl 0148.06701 · doi:10.1007/BF02416460
[21]Feng, C., ’On the existence of periodic solutions for a certain system of third order nonlinear differential equations’, Annals of Differential Equations 11(3), 1995, 264–269.
[22]Kalantarov, V. and Tiryaki, A., ’On the stability results for third order differential-operator equations’, Turkish Journal of Mathematics 21(2), 1997, 179–186.
[23]Meng, F. W., ’Ultimate boundedness results for a certain system of third order nonlinear differential equations’, Journal of Mathematical Analysis and Applications 177(2), 1993, 496–509. · Zbl 0783.34042 · doi:10.1006/jmaa.1993.1273
[24]Tiryaki, A., ’Boundedness and periodicity results for a certain system of third order nonlinear differential equations’, Indian Journal of Pure and Applied Mathematics 30(4), 1999, 361–372.
[25]Tunç, C., ’On the boundedness and periodicity of the solutions of a certain vector differential equation of third-order’, Applied Mathematics and Mechanics (English edn.) 20(2), 1999, 163–170.
[26]Tunç, C., ’Uniform ultimate boundedness of the solutions of third-order nonlinear differential equations’, Kuwait Journal of Science & Engineering 32(1), 2005, 39–48.
[27]Tunç, C., ’On the asymptotic behavior of solutions of certain third-order nonlinear differential equation’, Journal of Applied Mathematics and Stochastic Analysis 2005(1), 2005, 29–35. · Zbl 1077.34052 · doi:10.1155/JAMSA.2005.29